Find the value of tan ^{-1}left(tan frac{3 pi | vedclass

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(D) We need to evaluate $	an ^{-1}\left(	an \frac{3 \pi}{4}ight)$.The principal value branch of $	an ^{-1} x$ is $\left(-\frac{\pi}{2}, \frac{\pi

Vedclass · Accepted Answer

$\frac{-\pi}{4}$

Vedclass · Answer

(D) We need to evaluate $	an ^{-1}\left(	an \frac{3 \pi}{4}ight)$.The principal value branch of $	an ^{-1} x$ is $\left(-\frac{\pi}{2}, \frac{\pi}{2}ight)$.Since $\frac{3 \pi}{4} 
otin \left(-\frac{\pi}{2}, \frac{\pi}{2}ight)$,we use the property $	an(\pi - 	heta) = -	an 	heta$.Thus,$	an \frac{3 \pi}{4} = 	an \left(\pi - \frac{\pi}{4}ight) = -	an \frac{\pi}{4} = 	an \left(-\frac{\pi}{4}ight)$.Now,$	an ^{-1}\left(	an \frac{3 \pi}{4}ight) = 	an ^{-1}\left(	an \left(-\frac{\pi}{4}ight)ight)$.Since $-\frac{\pi}{4} \in \left(-\frac{\pi}{2}, \frac{\pi}{2}ight)$,we have $	an ^{-1}\left(	an \left(-\frac{\pi}{4}ight)ight) = -\frac{\pi}{4}$.

Find the value of $\tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)$.

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